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STAR SELECTION PRINCIPLES: a SURVEY 1. Introduction There Are Khayyam J. Math. 1 (2015), no. 1, 82{106 STAR SELECTION PRINCIPLES: A SURVEY LJUBISAˇ D.R. KOCINACˇ Communicated by H.R. Ebrahimi Vishki Abstract. We review selected results obtained in the last fifteen years on star selection principles in topology, an important subfield of the field of selection principles theory. The results which we discuss concern also uniform structures and, in particular, topological groups and their generalizations. 1. Introduction There are many results in the literature which show that a number of topolog- ical properties can be characterized by using the method of stars. In particular it is the case with many covering properties of topological spaces. The method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions. More infor- mation on star covering properties can be found in [17], [45]. We use here such a method in investigation of selection principles for topological and uniform spaces. Although Selection Principles Theory is a field of mathematics having a rich his- tory going back to the papers by Borel, Menger, Hurewicz, Rothberger, Seirpi´nski in 1920{1930's, a systematic investigation in this area rapidly increased and attracted a big number of mathematicians in the last two-three decades after Scheeper's paper [54]. Nowadays, this theory has deep connections with many branches of mathematics such as Set theory and General topology, Game theory, Ramsey theory, Function spaces and hyperspaces, Cardinal invariants, Dimension theory, Uniform structures, Topological groups and relatives, Karamata theory. Researchers working in this area have organized four international mathemati- cal forums called \Workshop on Coverings, Selections and Games in Topology". Date: Received: 29 November 2014; Accepted: 30 December 2014. 2010 Mathematics Subject Classification. Primary 54D20; Secondary 54A35, 54B20, 54E15, 54H10, 91A44. Key words and phrases. Star selection principles, ASSM, selectively (a), uniform selection principles. 82 STAR SELECTION PRINCIPLES 83 There are several survey papers about selection principles theory (see, for exam- ple, [33, 34, 53] and the paper [73] for open problems). Two basic ideas in this theory are simple and may be described by the following two schemes: Scheme 1: To a topological property P associate selectively P as follows: P: for each A there is a B such that ... selectivelyP: For each sequence hAn : n 2 Ni there is a sequence hBn : n 2 Ni such that ... Scheme 2: A and B are given collections, π is a procedure of selection. Apply π to A to arrive to B. For example, if P is compactness (for each open cover U of a space X there is a finite subcover V), then selectively P is defined as follows: for each sequence hUn; n 2 Ni of open covers of X there is a sequence hVn : n 2 Ni of finite sets with V ⊂ U , n 2 , and S V covers X. This property is called the Menger n n N n2N n property (see below). Many other selective versions of classical topological concepts have been defined in this way. Three classical selection principles defined in general forms in [54] are: Let A and B be sets consisting of families of subsets of an infinite set X. Then the following selection hypothesis are defined: Sfin(A; B): for each sequence hAn : n 2 Ni of elements of A there is a sequence hB : n 2 i of finite sets such that for each n, B ⊂ A , and S B 2 B. n N n n n2N n S1(A; B): for each sequence hAn : n 2 Ni of elements of A there is a sequence hbn : n 2 Ni such that for each n, bn 2 An, and fbn : n 2 Ng is an element of B. Ufin(A; B): for each sequence hAn : n 2 Ni of elements of A there is a sequence S hBn : n 2 Ni such that for each n, Bn is a finite subset of An and f Bn : n 2 Ng 2 B. In this paper we use the following notation for collections of covers of a topo- logical space X: •O is the collection of all open covers of X; • Ω is the collection of !-covers of X. An open cover U of X is said to be an !-cover if each finite subset of X is contained in a member of U and X2 = U; • Γ denotes the collection of γ-covers of X. An open cover U of X is said to be a γ-cover if each point of X does not belong to at most finitely many elements of U. Then: M: Sfin(O; O) is the Menger property [47], [25]; R: S1(O; O) is the Rothberger property [50]; H: Ufin(Γ; Γ) is the Hurewicz property [25] The paper is organized in the following way. Immediately after this intro- duction in Section 2 we give information about terminology and notation, and 84 LJ.D.R. KOCINACˇ also about known topological constructions we use in this paper. In Section 3 we discuss in details star selection principles in topological spaces. The next two sections are devoted to neighbourhood and absolute star selection proper- ties, two variations of the properties we considered in Section 3. In particular, in Subsection 5.2 we report results on selectively (a) spaces. In the second part of the paper we turn attention to appearance of star selection properties in special classes of topological structures: uniform and quasi-uniform spaces, and, espe- cially, in topological and paratopological groups. Each section contains some open problems which can motivate new researches for work in this field. 2. Definitions and terminology Throughout the paper \space" means \topological space". By N, Z, and R we denote the set of natural numbers, integers, and real numbers, respectively. The symbol ! denotes the set of nonnegative integers and also the first infinite ordinal, while !1 is the first uncountable ordinal. The cardinality of continuum is denoted by c, and CH denotes the Continuum Hypothesis. Most of undefined notations and terminology are as in [18]. If X is a space, K a collection of subsets of X, A a subset of X, and x 2 X, then St(A; K) is the union of all elements in K which meet A. We write St(x; K) instead of St(fxg; K). We recall known topological constructions which will be used in next sections without special mention. A. (The Baire space !!) Let !! be the set of all functions f : ! ! ! (in fact, the countable Tychonoff power of the discrete space D(!)). A natural pre-order ≺∗ on !! is defined by f ≺∗ g if and only if f(n) ≤ g(n) for all but finitely many n. A subset F of !! is said to be dominating if for each g 2 !! there is a function f 2 F such that g ≺∗ f. A subset F of !! is called bounded if there is an g 2 !! such that f ≺∗ g for each f 2 F . The symbol b (resp. d) denotes the least cardinality of an unbounded (resp. dominating) subset of !!. Another uncountable small cardinal characterized (by Bartoszy´nskiin 1987) in terms of subsets of !! is the cardinal cov(M), the covering number of the ideal of meager subsets of R: cov(M) = minfjF j : F ⊂ !! such that 8g 2 !! 9f 2 F with f(n) 6= g(n)8n 2 !g: Recommended literature concerning uncountable small cardinals is [16] and [75]. B. (Ψ-spaces) A family A of infinite subsets of N is called almost disjoint if the intersection of any two distinct sets in A is finite. Let A be an almost disjoint family. The symbol Ψ(A) denotes the space N [A with the following topology: all points of N are isolated; a basic neighborhood of a point A in A is of the form fAg [ (N n F ), where F is a finite subset of N. C. (Pixley-Roy space) For a space X, let PR(X) be the space of all nonempty finite subsets of X with the Pixley-Roy topology [15]: for A 2 PR(X) STAR SELECTION PRINCIPLES 85 and an open set U ⊂ X, let [A; U] = fB 2 PR(X): A ⊂ B ⊂ Ug; the family f[A; U]: A 2 PR(X);U open in Xg is a base for the Pixley-Roy topology. Obviously ffxg : x 2 Xg is closed and discrete in PR(X). Therefore, PR(X) is Lindel¨ofif and only if X is countable. It is known that (1) for a T1-space X, PR(X) is always zero-dimensional, Tychonoff and hereditarily metacompact, and (2) PR(X) is developable if and only if X is first-countable (see [15]). D. (Alexandroff duplicate) Let (X; τ) be a topological space. The Alexan- droff duplicate of X (see [18], [12]) is the set AD(X) := X × f0; 1g equipped with the following topology. For each U 2 τ let Ub = U × f0; 1g. Define a base for a topology on AD(X) by B = B0 [B1, where B0 is the family of all sets Ub n (F × f1g) ⊂ AD(X), with U 2 τ and F a finite subset of X, and B1 = fhx; 1i : x 2 Xg. For every x 2 X put τx = fU 2 τ : x 2 Ug and Bhx;0i = fUb n fhx; 1ig : U 2 τxg, and Bhx;1i = ffhx; 1igg. Then, if X is a T1-space, 0 S Bhx;0i is a local base at each hx; 0i 2 AD(X), and B = x2X (Bhx;0i [ Bhx;1i) is a base in AD(X) such that B0 ⊂ B.
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